
# q-expansion of newform 2268.2.l.m, downloaded from the LMFDB on 06 September 2025.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 2268
weight = 2
poly_data = [1, 1, 5, -4, 15, -2, 4, 0, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0, 0, 0, 0, 0, 0], 1], [[92, -1586, 437, -391, -14, -92, 23, 14], 289], [[-240, 305, -1140, 1020, -353, 240, -60, 64], 289], [[326, 1693, -1197, 2227, -414, 541, -63, 125], 289], [[-773, -854, -854, -3434, -52, -961, -121, -237], 289], [[-229, 1730, -1160, 4080, -562, 1096, 15, 273], 289], [[-444, -86, -86, -4471, 344, -1290, -111, -344], 289], [[-189, -970, 1920, -5916, 990, -1545, 25, -412], 289]]

hecke_ring_character_values = [[1135, [1, 0, 0, 0, 0, 0, 0, 0]], [1541, [0, 0, 1, 0, 0, 0, 0, 0]], [325, [-1, 0, -1, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1], [-1, 0, 0, -1, 0, 1, 1, 0], [0, 1, 1, 0, -1, 0, 0, 1], [0, 0, -1, 1, 0, 1, 0, 1], [-2, 1, -2, -2, -2, 2, 1, 2], [-1, 0, 0, 1, -1, -2, -1, -1], [-1, -2, -1, 2, 2, -2, -2, -2], [-1, -2, -1, 2, 1, -1, -2, -2], [1, -1, 1, 0, 1, -1, -1, 0], [0, 0, 0, 1, -1, 1, 0, 2], [-2, -2, -2, 1, 1, -1, -2, -1], [0, 2, -3, 0, -1, 0, 0, 1], [0, 1, 6, 2, 2, 2, 0, 0], [2, 1, 2, 0, -2, 2, 1, 0], [0, 2, 4, 1, 2, 1, 0, -1], [0, 5, 0, -1, -2, 2, 5, 1], [-1, 0, 0, 2, 1, -1, 2, 1], [0, -1, 3, -2, 0, -2, 0, -2], [0, 1, 1, 0, 2, 0, 0, -2], [10, 2, 10, -1, 0, 0, 2, 1], [-6, -3, -6, 4, 2, -2, -3, -4], [-3, 3, -3, 1, 1, -1, 3, -1], [-2, 0, 0, -1, 1, 2, 1, 1], [-4, 0, 0, 0, -1, -1, -4, -1], [3, 0, 3, 4, 2, -2, 0, -4], [0, 3, 6, -2, -2, -2, 0, 0], [0, 2, -2, 2, -1, 2, 0, 3], [-2, 0, 0, -1, -2, -1, -1, -2], [1, 0, 0, -1, -3, -2, 3, -3], [-5, 0, 0, 5, 1, -4, 1, 1], [0, 0, 9, 2, 3, 2, 0, -1], [3, 0, 0, 0, 1, 1, 4, 1], [9, 0, 0, -1, 0, 1, 6, 0], [-2, 0, -2, -3, -3, 3, 0, 3], [4, 2, 4, -2, -3, 3, 2, 2], [0, -2, 1, -1, -2, -1, 0, 1], [0, 0, -5, 1, 4, 1, 0, -3], [0, 3, 6, 2, -2, 2, 0, 4], [13, 0, 0, -1, 1, 2, 2, 1], [0, -2, 9, -3, 1, -3, 0, -4], [1, 1, 1, 5, 3, -3, 1, -5], [-1, 0, 0, 0, 1, 1, 3, 1], [0, 1, -4, 2, -1, 2, 0, 3], [0, -5, -7, -1, 2, -1, 0, -3], [-8, -3, -8, 0, 3, -3, -3, 0], [-9, 0, 0, 0, -1, -1, -4, -1], [6, 0, 0, -2, -1, 1, -2, -1], [7, -3, 7, -1, -1, 1, -3, 1], [0, 1, 6, -2, -3, -2, 0, 1], [8, 0, 0, 6, 1, -5, -2, 1], [18, 0, 0, -3, 0, 3, 0, 0], [6, 0, 0, 5, 4, -1, -3, 4], [-15, 0, 0, -5, -1, 4, 3, -1], [0, -4, -8, 0, 1, 0, 0, -1], [-1, -5, -1, 4, 2, -2, -5, -4], [8, 0, 0, -1, 0, 1, -1, 0], [-5, 6, -5, -3, -5, 5, 6, 3], [-3, 2, -3, -2, -1, 1, 2, 2], [0, -1, 9, 1, 7, 1, 0, -6], [6, 0, 0, 5, 2, -3, -4, 2], [-1, -6, -1, -1, 0, 0, -6, 1], [0, 0, 18, 1, 1, 1, 0, 0], [0, 4, 9, 3, 1, 3, 0, 2], [0, 2, -4, -3, 2, -3, 0, -5], [0, -4, 1, -1, 7, -1, 0, -8], [0, 3, 0, -3, 0, 0, 3, 3], [-12, 6, -12, -2, -5, 5, 6, 2], [0, 0, 0, 8, -1, -9, -5, -1], [5, 5, 5, -7, -5, 5, 5, 7], [-7, 0, 0, -3, -6, -3, -3, -6], [3, 0, 0, -8, -1, 7, 4, -1], [-6, 0, 0, 4, -2, -6, 0, -2], [5, 0, 0, -2, -1, 1, -1, -1], [-26, 0, 0, 0, 1, 1, -1, 1], [-13, 1, -13, 2, 1, -1, 1, -2], [0, 0, 0, -4, 4, 8, 3, 4], [-11, -1, -11, 6, 4, -4, -1, -6], [0, 0, -9, -2, -4, -2, 0, 2], [2, 1, 2, 5, -2, 2, 1, -5], [0, -9, 9, -4, 4, -4, 0, -8], [14, 0, 0, -4, -3, 1, 5, -3], [0, -8, -2, -4, 3, -4, 0, -7], [0, -6, -1, -1, -4, -1, 0, 3], [1, 0, 0, -6, -4, 2, 3, -4], [0, -11, -3, -2, 5, -2, 0, -7], [20, 3, 20, -8, -5, 5, 3, 8], [0, 7, 3, 4, 2, 4, 0, 2], [-19, 3, -19, -6, -7, 7, 3, 6], [2, 0, 0, -5, 0, 5, 12, 0], [0, -4, 17, 3, -1, 3, 0, 4], [0, -8, -12, -2, 0, -2, 0, -2], [-15, 0, 0, 2, 2, 0, -4, 2], [24, 0, 0, -4, -3, 1, -4, -3], [0, 0, 0, -1, 4, 5, -3, 4], [0, 2, -3, -7, -3, -7, 0, -4], [-14, 1, -14, 5, 3, -3, 1, -5], [-10, -5, -10, -1, 1, -1, -5, 1], [11, 4, 11, -5, 2, -2, 4, 5], [0, -4, 16, 0, 1, 0, 0, -1], [19, 2, 19, -3, -1, 1, 2, 3], [0, 1, 24, 3, 1, 3, 0, 2], [-19, -5, -19, 2, 1, -1, -5, -2], [0, 1, 6, 5, 1, 5, 0, 4], [31, 1, 31, 1, -1, 1, 1, -1], [-9, 0, -9, -5, -4, 4, 0, 5], [6, 7, 6, -2, 1, -1, 7, 2], [-17, 5, -17, -4, -1, 1, 5, 4], [7, 0, 0, -4, 1, 5, 8, 1], [0, 8, 4, 1, -7, 1, 0, 8], [0, -10, -25, 3, 6, 3, 0, -3], [3, 0, 0, -4, -2, 2, 4, -2], [-4, 0, 0, 3, 7, 4, -5, 7], [-19, 0, 0, 3, -3, -6, -4, -3], [0, -3, 2, -3, 0, -3, 0, -3], [0, 8, 12, -1, -12, -1, 0, 11], [15, 0, 0, 13, 4, -9, -10, 4], [-2, 10, -2, 1, 2, -2, 10, -1], [-6, 1, -6, -3, 1, -1, 1, 3], [19, 4, 19, 1, 1, -1, 4, -1], [0, -7, -21, 3, 8, 3, 0, -5], [0, -5, 3, -1, -4, -1, 0, 3], [0, -7, 11, 3, -4, 3, 0, 7], [-11, 0, 0, 0, 5, 5, -9, 5], [0, -2, -4, -1, -4, -1, 0, 3], [-9, -4, -9, 12, 10, -10, -4, -12], [-6, -8, -6, 0, 1, -1, -8, 0], [19, 0, 0, -1, -2, -1, 8, -2], [0, -1, -2, 2, 1, 2, 0, 1], [0, 8, 18, -2, -2, -2, 0, 0], [-2, 0, 0, -10, -5, 5, 5, -5], [-17, 0, 0, -2, -4, -2, -1, -4], [2, 0, 0, -4, -1, 3, 3, -1], [0, -2, 18, 1, -1, 1, 0, 2], [0, 3, -2, 3, 8, 3, 0, -5], [5, 1, 5, 10, 8, -8, 1, -10], [0, 1, 18, 1, 0, 1, 0, 1], [0, 1, 16, 0, -9, 0, 0, 9], [3, 0, 0, 2, -1, -3, -7, -1], [0, -5, 17, -7, 3, -7, 0, -10], [-15, -12, -15, 10, 10, -10, -12, -10], [-21, 0, 0, -7, -3, 4, 8, -3], [0, -8, 21, 1, 5, 1, 0, -4], [19, -2, 19, 4, 2, -2, -2, -4], [-20, -3, -20, 9, 6, -6, -3, -9], [-6, 0, 0, 1, -1, -2, 12, -1], [-1, 0, 0, -9, 1, 10, 13, 1], [5, 10, 5, 1, -3, 3, 10, -1], [3, 0, 0, 5, 3, -2, -3, 3], [3, 0, 0, -5, 0, 5, 4, 0], [26, 0, 0, -4, 0, 4, 8, 0], [15, 0, 0, 4, 1, -3, 2, 1], [-5, 0, 0, -10, -6, 4, 4, -6], [-12, -5, -12, -9, -7, 7, -5, 9], [6, 4, 6, 0, -5, 5, 4, 0], [0, 11, -6, 3, -10, 3, 0, 13], [26, 0, 0, -1, -7, -6, 4, -7], [-22, 5, -22, -5, -10, 10, 5, 5], [0, -12, -11, 7, 10, 7, 0, -3], [-4, 0, 0, 8, -2, -10, -4, -2], [0, 5, -10, -2, 1, -2, 0, -3], [-12, -6, -12, 7, 11, -11, -6, -7], [6, -1, 6, 5, -1, 1, -1, -5], [-15, 0, 0, 6, -1, -7, -7, -1], [0, 2, 14, -8, -5, -8, 0, -3], [-7, 0, 0, -6, -3, 3, 6, -3], [-10, 0, 0, 9, -1, -10, -7, -1], [-3, 0, 0, -2, -5, -3, -1, -5], [-23, 0, 0, 8, 1, -7, -5, 1], [0, -4, 24, -8, -3, -8, 0, -5], [-21, 0, 0, -8, -2, 6, 5, -2], [1, 0, 0, -7, 7, 14, 2, 7], [-7, -10, -7, 7, -2, 2, -10, -7], [0, 8, 7, 2, -10, 2, 0, 12], [-4, -2, -4, 8, 4, -4, -2, -8], [0, -1, 14, 7, 14, 7, 0, -7], [-4, 0, 0, -6, 4, 10, 4, 4], [0, 2, 38, 3, 5, 3, 0, -2], [0, -1, -11, 2, 13, 2, 0, -11], [10, 2, 10, -10, -6, 6, 2, 10], [0, -7, -12, -2, 3, -2, 0, -5], [-4, 9, -4, 0, -10, 10, 9, 0], [-22, -10, -22, 10, 2, -2, -10, -10], [9, 0, 0, 7, 5, -2, -9, 5], [0, -4, 13, 4, 14, 4, 0, -10], [11, 0, 0, 10, 0, -10, -5, 0], [13, 0, 0, 2, 0, -2, -2, 0], [0, 9, 33, 5, 7, 5, 0, -2], [-24, -13, -24, 7, 2, -2, -13, -7], [0, 8, 0, -2, 0, 0, 8, 2], [3, 0, 3, -14, -5, 5, 0, 14], [0, 3, -7, -10, -4, -10, 0, -6], [0, -6, 9, 8, 4, 8, 0, 4], [-2, 3, -2, 7, 3, -3, 3, -7], [4, 16, 4, -8, -11, 11, 16, 8], [0, 6, 3, 0, -10, 0, 0, 10], [9, 8, 9, 1, -9, 9, 8, -1], [-18, 12, -18, -14, -16, 16, 12, 14], [12, -4, 12, -4, -4, 4, -4, 4], [2, 11, 2, 1, -5, 5, 11, -1], [-13, 0, 0, 9, -2, -11, -17, -2], [10, 0, 0, -2, 2, 4, -10, 2], [15, 0, 0, 1, 0, -1, -5, 0], [0, 1, -15, 8, 5, 8, 0, 3], [10, -10, 10, 1, 9, -9, -10, -1], [-9, 0, 0, 6, 1, -5, 4, 1], [-12, -8, -12, 3, -4, 4, -8, -3], [34, 5, 34, -8, -3, 3, 5, 8], [-8, -4, -8, -6, -8, 8, -4, 6], [0, -4, -5, 2, -10, 2, 0, 12], [8, -8, 8, 1, 5, -5, -8, -1], [0, 8, 16, 10, 6, 10, 0, 4], [24, 4, 24, 4, 4, -4, 4, -4], [0, -1, -10, 7, 7, 7, 0, 0], [0, 7, 0, 6, 4, -4, 7, -6], [2, 0, 0, -8, -3, 5, -3, -3], [-14, -11, -14, 6, 3, -3, -11, -6], [0, -13, -7, 0, 3, 0, 0, -3], [-9, 0, 0, 0, -8, -8, -7, -8], [0, 2, -21, -2, -6, -2, 0, 4], [-6, 0, 0, -4, -5, -1, -3, -5], [15, -2, 15, 0, 4, -4, -2, 0], [0, -5, 3, 1, 9, 1, 0, -8], [31, -1, 31, -5, -3, 3, -1, 5], [0, 7, 11, -1, -6, -1, 0, 5], [0, 13, -8, 9, -6, 9, 0, 15], [0, 8, 10, -1, -14, -1, 0, 13], [0, 7, 6, -9, -5, -9, 0, -4], [-1, 4, -1, 5, 7, -7, 4, -5], [0, 14, -18, 1, -9, 1, 0, 10], [0, 7, 3, 1, -16, 1, 0, 17], [18, -4, 18, 8, 5, -5, -4, -8], [13, 14, 13, 0, -5, 5, 14, 0], [-24, 0, 0, 8, 5, -3, 4, 5], [9, 0, 0, 13, 8, -5, -14, 8], [12, 6, 12, 1, 2, -2, 6, -1], [0, -1, -6, 7, 13, 7, 0, -6], [13, 0, 0, -10, 5, 15, 2, 5], [-7, 0, 0, -1, 0, 1, 7, 0], [-14, 9, -14, -3, 6, -6, 9, 3], [-8, -3, -8, -12, -9, 9, -3, 12], [-34, -2, -34, 10, 5, -5, -2, -10], [0, -13, -10, 0, -3, 0, 0, 3], [0, -13, -15, 6, 14, 6, 0, -8], [1, 0, 0, 20, 0, -20, -2, 0], [43, 10, 43, 2, 1, -1, 10, -2], [34, 6, 34, 9, 6, -6, 6, -9], [0, 0, 0, 2, 0, -2, 17, 0], [0, -4, 21, 1, 6, 1, 0, -5], [24, -7, 24, -3, 7, -7, -7, 3], [-3, -3, -3, -3, 6, -6, -3, 3], [-36, 2, -36, -7, -8, 8, 2, 7], [23, 0, 0, -9, 0, 9, 8, 0], [-17, 5, -17, 7, 4, -4, 5, -7], [0, 2, -4, 1, 4, 1, 0, -3], [-29, 0, 0, -7, 1, 8, -1, 1], [-17, 0, 0, 13, 2, -11, -14, 2], [-5, 6, -5, 6, -6, 6, 6, -6], [-18, 0, 0, -5, -7, -2, 4, -7], [-37, -8, -37, 11, 2, -2, -8, -11], [-18, -13, -18, 5, 10, -10, -13, -5], [-9, 0, 0, 10, 1, -9, -12, 1], [-36, 9, -36, -6, -6, 6, 9, 6], [0, 0, 6, -2, 10, -2, 0, -12], [3, 0, 0, -18, -3, 15, 15, -3], [-4, 2, -4, -12, -9, 9, 2, 12], [0, -13, -27, -8, -6, -8, 0, -2], [-3, -6, -3, -3, -3, 3, -6, 3], [7, 0, 0, -10, 1, 11, 2, 1], [0, 1, 11, -1, 8, -1, 0, -9], [-5, 0, 0, -4, -5, -1, -16, -5], [14, 0, 0, 12, 5, -7, -4, 5], [0, -4, -9, 7, 15, 7, 0, -8], [44, -2, 44, 7, 5, -5, -2, -7], [-7, 0, 0, 3, 11, 8, -8, 11], [-11, 0, 0, -13, -14, -1, 5, -14], [3, -1, 3, 4, -1, 1, -1, -4], [0, 11, 30, -3, -10, -3, 0, 7], [3, -5, 3, -11, -5, 5, -5, 11], [-4, 10, -4, -13, -8, 8, 10, 13], [6, 14, 6, 1, -6, 6, 14, -1], [1, 4, 1, -4, -6, 6, 4, 4], [25, 0, 0, -11, 5, 16, 11, 5], [-38, -12, -38, -4, 5, -5, -12, 4], [0, 16, -8, 3, -19, 3, 0, 22], [0, 19, -22, 0, -10, 0, 0, 10], [29, 0, 0, -6, -7, -1, 14, -7], [-39, 0, 0, 8, -12, -20, -19, -12], [15, 0, 0, -8, -4, 4, -15, -4], [0, -7, 0, 5, -6, 5, 0, 11], [-6, 0, 0, -3, -2, 1, -11, -2], [0, 3, -23, -4, 6, -4, 0, -10], [-35, -6, -35, 3, 0, 0, -6, -3], [6, -5, 6, -8, 1, -1, -5, 8], [0, 18, -25, 7, -4, 7, 0, 11], [-6, 7, -6, -3, 2, -2, 7, 3], [9, -7, 9, 0, 7, -7, -7, 0], [47, 2, 47, -8, -5, 5, 2, 8], [-30, 0, 0, -1, 6, 7, 9, 6], [12, 16, 12, -10, -9, 9, 16, 10], [0, 11, 9, 4, -6, 4, 0, 10]]
